More about expanding products of binomials

I’ve been thinking about how to engage students in as many different senses as possible when learning math, so as to ensure that students have many distinct memories about a particular topic. The hope is that these different ways of knowing mathematics will create a more robust schema for the information that is acquired, which will then lead to better retention and deeper comprehension.

Ruling out the sense of smell (maybe “scratch and sniff” math books?) and taste, the remaining three senses that can be used to experience math are sight, sound and touch. Students hear math and see math frequently–that’s easy to do. Helping students experience mathematics using their hands or bodies usually requires a bit more creativity.

As I wrote in a previous post, I used Algebra Tiles and the “general area model” to help students calculate the product of two binomials using their visual and tactile senses. But since this approach can be problematic when working with negative quantities or higher powers of variables, it’s best to transition students to more general ways of knowing mathematics as they develop greater skill and understanding.

The ultimate goal is to get students to think of expanding products of polynomials as a generalization of the simple distributive property:  a(b+c) = ab + ac. (See note below about “FOIL.”) But how to get students to experience this visually or kinesthetically?

Here’s a silly thing that I did in my Algebra 1 class today: After showing students the “general area model” for expanding products of binomials, I asked two pairs of students to come up to the front of the room. The students’ names start with the letters A, S, E, and G. I purposely chose these students because A and S are close friends, as are E and G, but the two pairs don’t seem to associate with each other much. I wrote on the board (a + s)(e +g), then asked the four students to help me act out a short scene.

The scene: A and S, who are close friends, are going to a party hosted by E and G, who are also close friends. A and S walk in the door and are greeted by E and G. All four people shake hands and introduce themselves and each other.

I then let the students act out the scene in front of the whole class. Some students were shy, which made the whole thing a bit awkward, but they managed to do it just fine. I then asked the class what they saw–in particular, who shook hands with whom?

The students easily pointed out that A shook hands with E and G, then S shook hands with E and G. They also could explain to me that the reason A and S don’t shake hands is that they know each other already (which is the same reason why E and G don’t shake hands).

The point of all of this was to help students kinesthetically experience this product.

A portion of students' notes from class today. (Students had to copy down the handwritten parts.)

Each of the four terms in the answer matches up with one of four handshakes. The diagram (with colored arrows and terms) then became the visual reinforcement for the kinesthetic experience. Maybe a bit hokey, but I think this short demonstration made the idea of using the distributive law very concrete for some students.

Teachers: I’d love to hear what other activities you use to help students engage in mathematics using other senses. One of my close friends gets students to walk and run while graphing their motion, to help students learn the concept of slope.

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Footnote about “FOIL”: The “FOIL” acronym is often taught to students to help them remember how to expand products of two binomials.

(x+2)(x+3) = FIRST (x^2) + OUTER (3x) + INNER (2x) + LAST (6)

It’s a handy way to remember things, for sure, but I am consciously trying not to teach this to my students because it only applies to the product of two binomials–it doesn’t help if you are multiplying a polynomials with more than two terms, for example, (x+2)(x+y+3). It will be interesting to see if students can more easily make the jump to expanding (x+2)(x+y+3) because I’ve consciously avoided teaching  “FOIL” and instead focused on this idea of generalizing the distributive property.

4 thoughts on “More about expanding products of binomials

  1. For graphing in physics we use motion detectors that automatically plot distance vs time. Students can try different patterns of speed to see how that changes the slope.

    I’ve seen the distributive property taught with m&ms. I can’t remember all the details, but I think for 2(4 6) there was a person with a bag of 4 m&ms and a person with a bag of 6. The “distributor” student went to each person and doubled their m&ms.

  2. A general term I’ve heard used over FOIL is EWE, Each With Each. In your example, it would be each of A & S shakes hands with each of E & G. It avoids the binomial * binomial specificity, but has some cons: I’d guess the word “ewe” is less familiar than “foil,” and the “each with each” mnemonic alone doesn’t explain why there are not AS and EG terms (whereas FOIL explicitly names which terms are multiplied). Anyone have any sense of whether EWE is catching on?

    • EWE is pretty recent, and isn’t catchy at all. I don’t see this catching on with teachers or students.

      This was a great activity Darryl! In a later opportunity (I am writing this far too late in the game) you could have 2 students in one group and 3 in the other and have the same kind of game, then the same kind of drawing.

      The only thing that troubles me, and it’s a very slight thing, is why this concept applies to multiplication and not some other operation too. The area model supports this well though, and I think it’s pretty unlikely a kid would have brought this up! But it might lead to some misunderstanding of (a+s) / (e+g)… not really a big problem.

  3. I love this idea of acting out multiplying polynomials. It makes so much sense when you do it this way! I am gong to try this with my students this coming year! Thanks for the hands on idea!

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